Differentiate the function.$ f(x) = 2^{40} $
Find $ \frac {d^9}{dx^9}(x^8 \ln x). $
Assume that all the given functions are differentiable.
If $ z = f(x, y) $, where $ x = r \cos \theta $ and $ y = r \sin \theta $, (a) find $ \partial z/ \partial r $ and $ \partial z/ \partial \theta $ and (b) show that $$ \biggl( \dfrac{\partial z}{\partial x} \biggr)^2 + \biggl( \dfrac{\partial z}{\partial y} \biggr)^2 = \biggl( \dfrac{\partial z}{\partial r} \biggr)^2 + \dfrac{1}{r^2}\biggl( \dfrac{\partial z}{\partial \theta} \biggr)^2 $$
Find the indicated partial derivative(s). $$u=e^{r \theta} \sin \theta ; \quad \frac{\partial^{3} u}{\partial r^{2} \partial \theta}$$
Finding an Indefinite Integral In Exercises 15- 36 , find the indefinite integral and check the result by differentiation.$\int \frac{x^{4}-3 x^{2}+5}{x^{4}} d x$
A power line is to be constructed from a power station at point A to an island at point C, which is 3 miles directly out in the water from a point B on the shore. Point B is 6 miles downshore from the power station at A. It costs $4200 per mile to lay the power line underwater and $3000 per mile to lay the line underground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A. (Hint: The length of CS is sqrt(9+x^2).)
In your backyard you would like to build an enclosed rectangulargarden that has an area of 72 square feet and faces yourhouse. Basic fencing costs $2 per foot which you plan on usingfor three sides of the garden. The last side, which faces yourhouse, you decide to use a more expensive fence that costs$6 per foot. The minimum cost (rounded) for your fence is
A water trough is being built out of sheet metal. The troughwill be a half-cylinder, as shown below. It must hold 30 cubic feetof water when full. What dimensions (r and h) should be used forthe trough in order to minimize the amount (in square feet) ofsheet metal required? For full credit, you must justify that youhave minimized, rather than maximized, the amount of sheet metalneeded.
Suppose that there are two very large reservoirs of water, one at a temperature of 91.0 °C and one at a temperature of 21.0 °C. These reservoirs are brought into thermal contact long enough for 38590 J of heat to flow from the hot water to the cold water. Assume that the reservoirs are large enough so that the temperatures do not change significantly.What is the total change in entropy, ΔS_tot, resulting from this heat exchange between the hot water and the cold water?ΔS_tot = J/K Calculate the amount of energy made unavailable for work by this increase in entropy. amount of energy unavailable for work: JHow much work could a Carnot engine do if it took in the given amount of heat (38590 J) from the hot water reservoir and exhausted heat to the cold water reservoir? work done by a Carnot engine: J
Consider the function, f(x) = 2x^3 - 1/2 x^2 - 2x(a) Find the critical numbers of the function.(b) Use a labeled sign chart to find the intervals where the function is increasing/decreasing.(c) Perform the First Derivative Test to find the relative extrema.
Using your knowledge of how two capacitors in series combine to give an equivalent capacitance (equation one) and also how two capacitors in parallel combine to give an equivalent capacitance (equation two), derive an equation for either C1 or C2.Because these are two equations in two unknowns, it is possible to do this. You will need to algebraically manipulate the two equations until you can use the quadratic formula. The two solutions to the quadratic equation (no matter whether you solve for C1 or C2) are the two capacitances you are looking for.Question: Submit your derivation of the equation that is then solved using the quadratic equation. This derivation should be algebraic (no numbers) and should only be written in terms of C1 or C2 and series Cequivalent and parallel Cequivalent. Make sure all your work is shown and that the final equation is clearly expressed.
